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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.NumberTheory.PiExponent.Ampleness.ClosedAmpleRestriction
import OAI.NumberTheory.PiExponent.Polynomials.PullbackFrameCoefficient
namespace OAI
noncomputable section
namespace PiExponent.CompositePullbackFrame
open AlgebraicGeometry CategoryTheory
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
variable {X Y Z : Scheme.{0}}
private theorem mapIso_trans_inv_comp {C D : Type*} [Category C] [Category D]
(F : C ⥤ D) {A B T : C} {U : D} (e : A ≅ B) (u : F.obj B ≅ U)
(s : A ⟶ T) :
(F.mapIso e ≪≫ u).inv ≫ F.map s = u.inv ≫ F.map (e.inv ≫ s) := by
simp only [Iso.trans_inv, Functor.mapIso_inv, Functor.map_comp, Category.assoc]
private theorem map_comp_iso_inv {C D : Type*} [Category C] [Category D]
(F : C ⥤ D) {A B T : C} {U : D} (u : U ⟶ F.obj A)
(s : A ⟶ B) (e : B ≅ T) :
(u ≫ F.map (s ≫ e.hom)) ≫ (F.mapIso e).inv = u ≫ F.map s := by
simp only [Functor.mapIso_inv, Category.assoc, ← Functor.map_comp,
Iso.hom_inv_id, Category.comp_id]
theorem exists_frame (f : X ⟶ Y) (φ : Z ⟶ X) [IsOpenImmersion φ]
(V : Y.Opens) (g : Z ⟶ V.toScheme) (h : φ ≫ f = g ≫ V.ι)
{M : Y.Modules} (e : M.restrict V.ι ≅ O V.toScheme) :
∃ E : ((Scheme.Modules.pullback f).obj M).restrict φ ≅ O Z,
∀ s : O Y ⟶ M,
coefficient E (restrictSection φ (pullbackSection f s)) =
g.appTop (coefficient e (restrictSection V.ι s)) := by
let F := Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor φ
let G := Scheme.Modules.restrictFunctor V.ι ⋙ Scheme.Modules.pullback g
let v : F ≅ G :=
Functor.isoWhiskerLeft (Scheme.Modules.pullback f)
(Scheme.Modules.restrictFunctorIsoPullback φ) ≪≫
Scheme.Modules.pullbackComp φ f ≪≫
Scheme.Modules.pullbackCongr h ≪≫
(Scheme.Modules.pullbackComp g V.ι).symm ≪≫
Functor.isoWhiskerRight (Scheme.Modules.restrictFunctorIsoPullback V.ι).symm
(Scheme.Modules.pullback g)
let uF : F.obj (O Y) ≅ O Z :=
(Scheme.Modules.restrictFunctor φ).mapIso (pullbackUnitIso f) ≪≫
Scheme.Modules.restrictUnitIso φ
let uG : G.obj (O Y) ≅ O Z :=
(Scheme.Modules.pullback g).mapIso (Scheme.Modules.restrictUnitIso V.ι) ≪≫
pullbackUnitIso g
obtain ⟨E, hE⟩ := PullbackFrameCoefficient.exists_frame_natIso_all F G v uF uG
(pullbackFrame g e)
refine ⟨E, fun s => ?_⟩
have hs : coefficient E (restrictSection φ (pullbackSection f s)) =
coefficient (pullbackFrame g e) (pullbackSection g (restrictSection V.ι s)) := by
simp only [restrictSection, pullbackSection]
exact (congrArg (coefficient E)
(mapIso_trans_inv_comp (Scheme.Modules.restrictFunctor φ)
(pullbackUnitIso f) (Scheme.Modules.restrictUnitIso φ)
((Scheme.Modules.pullback f).map s)).symm).trans
((hE s).trans (congrArg (coefficient (pullbackFrame g e))
(mapIso_trans_inv_comp (Scheme.Modules.pullback g)
(Scheme.Modules.restrictUnitIso V.ι) (pullbackUnitIso g)
((Scheme.Modules.restrictFunctor V.ι).map s))))
rw [hs, coefficient_pullback]
theorem exists_power_frame (L : LineBundle Y) (f : X ⟶ Y)
(φ : Z ⟶ X) [IsOpenImmersion φ] (V : Y.Opens)
(g : Z ⟶ V.toScheme) (h : φ ≫ f = g ≫ V.ι) (n : ℕ)
(e : (modulePow Y L.sheaf n).restrict V.ι ≅ O V.toScheme) :
∃ E : (modulePow X (L.pullback f).sheaf n).restrict φ ≅ O Z,
∀ s : O Y ⟶ modulePow Y L.sheaf n,
coefficient E (restrictSection φ (pullbackPowerSection L f n s)) =
g.appTop (coefficient e (restrictSection V.ι s)) := by
obtain ⟨E, hE⟩ := exists_frame f φ V g h e
refine ⟨(Scheme.Modules.restrictFunctor φ).mapIso
(PiExponentSeshadri.PullbackTensor.powIso f L n).symm ≪≫ E, fun s => ?_⟩
erw [coefficient_transport]
have hs : restrictSection φ (pullbackPowerSection L f n s) ≫
((Scheme.Modules.restrictFunctor φ).mapIso (PiExponentSeshadri.PullbackTensor.powIso f L n).symm).hom =
restrictSection φ (pullbackSection f s) := by
simp only [restrictSection, pullbackPowerSection, Functor.mapIso_hom, Iso.symm_hom]
exact map_comp_iso_inv (Scheme.Modules.restrictFunctor φ)
(Scheme.Modules.restrictUnitIso φ).inv (pullbackSection f s)
(PiExponentSeshadri.PullbackTensor.powIso f L n)
exact (congrArg (coefficient E) hs).trans (hE s)
end PiExponent.CompositePullbackFrame
end
end OAI
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